On Horn’s problem and its volume function. (English) Zbl 1487.22016
Summary: We consider an extended version of Horn’s problem: given two orbits \({{\mathcal{O}}}_\alpha\) and \({{\mathcal{O}}}_\beta\) of a linear representation of a compact Lie group, let \(A\in{\mathcal O}_\alpha , B\in{{\mathcal{O}}}_\beta\) be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum \(A+B\). We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of \(\text{SO}(n), \text{SU}(n)\) and \(\text{USp}(n)\) respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood-Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of \(B_2=\mathfrak{so}(5)\).
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
References:
| [1] | Berenstein, A.; Zelevinsky, A., Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys., 5, 453-472 (1988) · Zbl 0712.17006 · doi:10.1016/0393-0440(88)90033-2 |
| [2] | Berenstein, A.; Zelevinsky, A., Triple multiplicities for \(s\ell (r+1)\) and the spectrum of the exterior algebra of the adjoint representation, J. Algebr. Combin., 1, 7-22 (1992) · Zbl 0799.17005 · doi:10.1023/A:1022429213282 |
| [3] | Berenstein, A.; Zelevinsky, A., Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., 143, 77-128 (2001) · Zbl 1061.17006 · doi:10.1007/s002220000102 |
| [4] | Capparelli, S., Calcolo della funzione di partizione di Kostant, Bolletino dell’Unione Matematica Italiana, 8, 89-110 (2003) |
| [5] | Coquereaux, R.: Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups. arXiv:1209.6621 · Zbl 1295.81091 |
| [6] | Coquereaux, R.; Zuber, J-B, From orbital measures to Littlewood-Richardson coefficients and hive polytopes, Ann. Inst. Henri Poincaré Comb. Phys. Interact., 5, 339-386 (2018) · Zbl 1429.17009 · doi:10.4171/AIHPD/57 |
| [7] | Coquereaux, R., Zuber, J.-B.: The Horn problem for real symmetric and quaternionic self-dual matrices. SIGMA15, 029 (2019). http://www.emis.de/journals/SIGMA/2019/029/sigma19-029.pdf, arXiv:1809.03394 · Zbl 1451.15008 |
| [8] | Coquereaux, R.; Zuber, J-B, Conjugation properties of tensor product multiplicities, J. Phys. A: Math. Theor., 47, 455202 (2014) · Zbl 1327.14260 · doi:10.1088/1751-8113/47/45/455202 |
| [9] | De Loera, JA; McAllister, TB, On the computation of Clebsch-Gordan coefficients and the dilation effect, Exp. Math., 15, 7-19 (2000) · Zbl 1115.17005 · doi:10.1080/10586458.2006.10128948 |
| [10] | Derksen, H.; Weyman, J., On the Littlewood-Richardson polynomials, J. Algebra, 255, 247-257 (2002) · Zbl 1018.16012 · doi:10.1016/S0021-8693(02)00125-4 |
| [11] | Dooley, AH; Repka, J.; Wildberger, N., Sums of adjoint orbits, Linear Multilinear Algebra, 36, 79-101 (1993) · Zbl 0797.15010 · doi:10.1080/03081089308818278 |
| [12] | Duistermaat, JJ; Heckman, GJ, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., 69, 259-268 (1982) · Zbl 0503.58015 · doi:10.1007/BF01399506 |
| [13] | Etingof, P., Rains, E.: Mittag-Leffler type sums associated with root systems. arXiv:1811.05293 · Zbl 1357.13016 |
| [14] | Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Am. Math. Soc., 37, 209-249 (2000) · Zbl 0994.15021 · doi:10.1090/S0273-0979-00-00865-X |
| [15] | Guillemin, V.; Lerman, E.; Sternberg, S., Symplectic Fibrations and Multiplicity Diagrams (1996), Cambridge: Cambridge Unversity Press, Cambridge · Zbl 0870.58023 |
| [16] | Heckman, GJ, Projections of orbits and asymptotic behavior of multiplicities for compact connected lie groups, Invent. Math., 67, 333-356 (1982) · Zbl 0497.22006 · doi:10.1007/BF01393821 |
| [17] | Horn, A., Eigenvalues of sums of Hermitian matrices, Pac. J. Math., 12, 225-241 (1962) · Zbl 0112.01501 · doi:10.2140/pjm.1962.12.225 |
| [18] | Kirillov, AA, Lectures on the Orbit Method (2004), Providence: American Mathematical Society, Providence · Zbl 1229.22003 |
| [19] | Klyachko, AA, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), 4, 419-445 (1998) · Zbl 0915.14010 · doi:10.1007/s000290050037 |
| [20] | Knutson, A., The symplectic and algebraic geometry of Horn’s problem, Linear Algebra Appl., 319, 61-81 (2000) · Zbl 0981.15010 · doi:10.1016/S0024-3795(00)00220-2 |
| [21] | Knutson, A.; Tao, T., Honeycombs and sums of Hermitian matrices, Notices Am. Math. Soc., 48, 175-186 (2001) · Zbl 1047.15006 |
| [22] | Knutson, A.; Tao, T., The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, J. Am. Math. Soc., 12, 1055-1090 (1999) · Zbl 0944.05097 · doi:10.1090/S0894-0347-99-00299-4 |
| [23] | Knutson, A.; Tao, T.; Woodward, C., The honeycomb model of GL(n) tensor products II: puzzles determine facets of the Littlewood-Richardson cone, J. Am. Math. Soc., 17, 19-48 (2004) · Zbl 1043.05111 · doi:10.1090/S0894-0347-03-00441-7 |
| [24] | Lee, J., Smooth Manifolds (2003), New York: Springer, New York |
| [25] | Lie Package, van Leeuwen, M.A.A., Cohen, A.M., Lisser, B.: Computer Algebra Nederland, Amsterdam. ISBN 90-74116-02-7 (1992) |
| [26] | Macdonald, IG, Some conjectures for root systems, SIAM J. Math. Anal., 13, 988-1007 (1982) · Zbl 0498.17006 · doi:10.1137/0513070 |
| [27] | Opdam, EM, Some applications of hypergeometric shift operators, Invent. Math., 98, 1-18 (1989) · Zbl 0696.33006 · doi:10.1007/BF01388841 |
| [28] | Mathematica, Wolfram Research, Inc., Champaign, IL (2012). http://www.wolfram.com/ |
| [29] | McMullen, P.: Lattice invariant valuations on rational polytopes. Arch. Math. (Basel) 31, 509-516 (1978/1979) · Zbl 0387.52007 |
| [30] | McSwiggen, C., A new proof of Harish-Chandra’s integral formula, Commun. Math. Phys., 365, 239-253 (2019) · Zbl 1407.22006 · doi:10.1007/s00220-018-3259-9 |
| [31] | Mehta, ML; Dyson, FJ, Statistical theory of the energy levels of complex systems. V, J. Math. Phys, 4, 713-719 (1963) · Zbl 0133.45202 · doi:10.1063/1.1704009 |
| [32] | Pak, I.; Vallejo, E., Combinatorics and geometry of Littlewood-Richardson cones, Eur. J. Comb., 6, 995-1008 (2005) · Zbl 1063.05133 · doi:10.1016/j.ejc.2004.06.008 |
| [33] | Racah, G.: In: Gürsey, F. (ed.) Group Theoretical Concepts and Methods in Elementary Particle Physics, p. 1. Gordon and Breach, New York; D. Speiser, ibid., p. 237 (1964) |
| [34] | Shiga, K., Some aspects of real-analytic manifolds and differentiable manifolds, J. Math. Soc. Jpn., 16, 128-142 (1964) · Zbl 0137.42603 · doi:10.2969/jmsj/01620128 |
| [35] | Stanley, RP, Enumerative Combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0889.05001 |
| [36] | Steinberg, R., A general Clebsch-Gordan theorem, Bull. AMS, 67, 406-407 (1961) · Zbl 0115.25605 · doi:10.1090/S0002-9904-1961-10644-7 |
| [37] | Tarski, J., Partition function for certain simple Lie algebras, J. Math. Phys., 4, 569-574 (1963) · Zbl 0116.02401 · doi:10.1063/1.1703992 |
| [38] | Vergne, M.: private communication |
| [39] | Zuber, J-B, Horn’s problem and Harish-Chandra’s integrals. Probability distribution functions, Ann. Inst. Henri Poincaré Comb. Phys. Interact., 5, 309-338 (2018) · Zbl 1397.15008 · doi:10.4171/AIHPD/56 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
